Friday, February 28, 2014

What
is the negation of self-doubt? It would seem to be self-trust, for self-doubt
asserts that ‘this’ sentence is unprovable, while self-trust asserts that
‘this’ sentence is provable; but it’s a different ‘this’ for each one.
Self-reference implies a twist in logic. Paradox pervades this study.

In
fact the negation of self-doubt is not self-trust but self-shame; and the
negation of self-trust is not self-doubt but self-pride.

Thursday, February 27, 2014

Four
quanta are of particular interest to logicians. Here I nickname them the quanta
of Self-Doubt, Self-Shame, Self-Pride and Self-Trust.

The
quantum of Self-Doubt says

“
‘Is unprovable when quined’, is unprovable when quined.”

Or
in other words: “This sentence is not provable.”

Or
in other words:“Doubt me.”

It
is the mathematical quantum of uncertainty. Its equation is:

D=not
prv D

The
quantum of Self-Shame says

“
‘Is refutable when quined’, is refutable when quined.”

Or
in other words: “This sentence is provably false.”

Or
in other words:“Refute me.”

It
is the mathematical quantum of error. Its equation is:

S=prv
not S

The
quantum of Self-Pride says

“
‘Is irrefutable when quined’, is irrefutable when quined.”

Or
in other words: “This sentence is possible.”

Or
in other words:“Tolerate me.”

It
is the mathematical quantum of power. Its equation is:

P=not
prv not P

The
quantum of Self-Trust says

“
‘Is provable when quined’, is provable when quined.”

Or
in other words: “This sentence is provable.”

Or
in other words:“Trust me.”

It
is the mathematical quantum of certainty. Its equation is:

T=prv
T

The
quantum of self-doubt is also known as a Gödelian sentence. It is literally a
paradox, for it calls itself beyond belief. The quantum of self-trust is also
known as a Henkin sentence (after the man who asked if it is true) or a Löbian
sentence (after the man who proved that it is). The quanta of self-shame and
self-pride are unclaimed, for good reason.

Wednesday, February 26, 2014

Sentences
can be defined in terms of each other; can a sentence be defined in terms of
itself? Yes! Self-reference is possible, even in a rigorously hierarchical
logic system, due to a technical trick called ‘quining’. To ‘quine’ a predicate
means to apply it to its own quotation, For instance:

‘Is
a predicate’ is a predicate.

‘Is
not a predicate’ is not a predicate.

‘Is
a statement when quined’ is a statement when quined.

The
first and the third are true, the second is false. The third is self-referential;
when the quoted phrase is quined, the result is the original sentence. In
general, the statement

“Has
property F when quined’ has property F when quined.

is
self-referential; it says that it has property F:

“This
statement has property F.”

S=F(S)

Statement
S generates itself out of itself. It is a self-propagating process; an organic
structure. I call it a ‘logical quantum’.

Quanta
bootstrap themselves into definition. Like the Earth that we stand on, they
rest upon themselves. Quanta need no prior ‘foundation’, any more than our
round planet, afloat in the void, needs to lie on the back of a space turtle.

Tuesday, February 25, 2014

Every
proof uses its own forms of notation, which has its own conveniences and hidden
assumptions. I set forth this paper’s notation here in order to simplify
critique. To quote the famous logician Humpty Dumpty, “when I use a word, it
means just what I choose it to mean – neither more nor less.”

This
paper discusses mathematical propositions, called ‘sentences’. They are defined
in terms of each other, using logical connectives such as ‘and’, ‘or’ and
‘no’.Call logically equivalent
sentences ‘equal’; denote equality by ‘=’. Let T denote anything obviously
true, and F denote anything obviously false. Therefore T = not F; the former
can mean ‘x=x’ or ‘1+1=2’ or ‘0 does not equal 1’; and F denotes the denial of
those statements.

Some
sentences are provable. The assertion that the sentence ‘S’ is provable is
another sentence; call it “prv S”. To assert prv S is to assert that S is necessarily
true, therefore true in all mathematical models.

Some
sentences are true on occasion. Such sentences are ‘possible’. The assertion
that the sentence ‘S’ is possible is another sentence; call it “poss S”. To
assert poss S is to assert that S is true in some mathematical model; and
therefore it cannot be proven false. Therefore poss and prv are conjugate to
each other:

poss
S=not prv not S

prv
S=not poss not S

The
possible is what you can’t prove false; and the provable is what you can’t
possibly deny. We can form combinations of these:

Poss
T=‘truth is possible’

Prv
F=‘falsehood is provable’

Poss
F=‘falsehood is possible’

Prv
T=‘truth is provable’

The
last two are easy to simplify. T is provable in all mathematical models, in
fact it is the standard of proof; therefore prv T = T. Similarly poss F is
false in all mathematical models, therefore poss F = F.

The
statements poss T and prv F involve a deep conundrum; namely, is our proof
system itself valid? Perhaps our reasoning methods contain fatal flaws. Is our
logic in fact consistent? Does it have a mathematical model? If so, then truth
is possible; if not, then falsehood is provable.

This
paper is about the mathematical logic of belief systems. Idiscuss four forms of self-reference, and
their paradoxical properties. These imply Gödel’s Incompleteness Theorems,
which in turn imply Löb’s Theorem.

Löb’s
Theorem says that any statement that asserts just its own provability is, in
fact, provable. It is a logical bootstrap; by declaring itself necessary, it
makes itself necessary. A Löbian statement, by its perfect faith in itself,
attains truth.

But
why? How can anything so vain as self-belief attain absolute certainty? Read
on, and I shall prove it to you.